📚 Year 12 CCEA Mathematics: Formula & Theorem Quick Reference Handbook | 十二年级 CCEA 数学公式定理速查手册
This quick reference handbook covers the essential formulas, theorems and key results required for the CCEA Year 12 AS Mathematics course. It is designed as a compact revision aid for students working through Pure Mathematics, Mechanics and Statistics. Each section pairs an English statement of a result with its Chinese translation, ensuring clarity for bilingual learners. Use this handbook to check your recall of critical facts before tests and the final examination.
这本速查手册涵盖了 CCEA 十二年级 AS 数学课程所需要的主要公式、定理和关键结论。它是一份紧凑的复习工具，适用于纯数学、力学和统计学的学习。每一节都给出一个结论的英文表述并配以中文译文，确保双语学习者能清晰理解。在单元测验和最终考试前，可用本手册快速核查重要知识点的掌握情况。
1. Algebra Basics | 代数基础
Quadratic formula: x = [−b ± √(b² − 4ac)] / 2a
For any quadratic equation ax² + bx + c = 0, the solutions are given by the quadratic formula above.
对于任意二次方程 ax² + bx + c = 0，其解由上述二次公式给出。
Laws of indices: aᵐ × aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1, a⁻ⁿ = 1/aⁿ
These index laws hold for any real numbers a > 0 and rational exponents m, n.
以上指数定律适用于任意实数 a > 0 以及有理指数 m, n。
Remainder theorem: When a polynomial f(x) is divided by (x − a), the remainder is f(a).
If f(a) = 0, then (x − a) is a factor of f(x) (factor theorem).
若 f(a) = 0，则 (x − a) 是 f(x) 的一个因式（因式定理）。
2. Coordinate Geometry | 坐标几何
Distance between two points: d = √[(x₂ − x₁)² + (y₂ − y₁)²]
The length of the line segment joining (x₁, y₁) and (x₂, y₂) is given by Pythagoras’ theorem.
连接点 (x₁, y₁) 和 (x₂, y₂) 的线段长度由勾股定理给出。
Midpoint: M = ((x₁+x₂)/2, (y₁+y₂)/2)
The midpoint coordinates are the averages of the x- and y-coordinates.
中点坐标是两个对应坐标的算术平均值。
Equation of a straight line: y − y₁ = m(x − x₁), where m = (y₂ − y₁)/(x₂ − x₁)
The gradient m measures the steepness and direction of the line. Parallel lines have equal gradients; perpendicular lines satisfy m₁ × m₂ = −1.
斜率 m 表示直线的倾斜程度和方向。平行直线的斜率相等；垂直直线的斜率满足 m₁ × m₂ = −1。
Equation of a circle: (x − a)² + (y − b)² = r²
The circle has centre (a, b) and radius r. Completing the square rewrites a general circle equation into this standard form.
该圆的圆心为 (a, b)，半径为 r。通过配方法可将一般式转化为标准式。
3. Trigonometry | 三角学
Exact values: sin30°=½, sin45°=√2/2, sin60°=√3/2; cos30°=√3/2, cos45°=√2/2, cos60°=½; tan30°=1/√3, tan45°=1, tan60°=√3
These exact trigonometric ratios for standard angles must be memorised.
应熟记这些特殊角度的精确三角比值。
Sine rule: a/sin A = b/sin B = c/sin C
Used for non‑right‑angled triangles when you know either two angles and a side, or two sides and a non‑included angle.
该公式用于非直角三角形，已知两角一边或两边及一个非夹角的情况。
Cosine rule: a² = b² + c² − 2bc cos A
Used when you know two sides and the included angle, or all three sides, to find an unknown side or angle.
当已知两边及其夹角或三边时，可用此公式求未知边或角。
Area of a triangle: Area = ½ ab sin C
The area is half the product of two sides and the sine of the included angle.
三角形面积等于两边长与其夹角正弦的乘积的一半。
Pythagorean identity: sin²θ + cos²θ = 1
This fundamental identity links sine and cosine for any angle θ.
这一基本恒等式适用于任意角 θ。
4. Sequences and Series | 数列与级数
Arithmetic sequence: nth term uₙ = a + (n − 1)d
Here a is the first term and d is the common difference.
其中 a 为首项，d 为公差。
Arithmetic series sum: Sₙ = n/2 [2a + (n − 1)d] = n/2 (a + l)
The sum of the first n terms can be expressed using either the first term and common difference, or the first and last term l.
前 n 项的和可用首项与公差，或用首项与末项 l 表示。
Geometric sequence: nth term uₙ = arⁿ⁻¹
Here a is the first term and r is the common ratio (r ≠ 0).
其中 a 为首项，r 为公比（r ≠ 0）。
Geometric series sum: Sₙ = a(1 − rⁿ)/(1 − r) for r ≠ 1
This formula gives the sum of the first n terms of a geometric progression.
该公式给出了等比数列前 n 项的和。
Sum to infinity: S∞ = a/(1 − r), valid for |r| < 1
When the common ratio’s absolute value is less than 1, the infinite geometric series converges to a finite sum.
当公比的绝对值小于 1 时，无穷等比级数收敛到一个有限和。
5. Exponentials and Logarithms | 指数与对数
Definition: y = logₐ x ⟺ aʸ = x, where a > 0, a ≠ 1
The logarithm is the inverse of the exponential function with the same base.
对数是指数函数的反函数，两者底数相同。
Laws of logarithms: logₐ(xy) = logₐ x + logₐ y; logₐ(x/y) = logₐ x − logₐ y; logₐ(xⁿ) = n logₐ x
These rules hold for any positive x, y.
上述运算法则适用于任意正数 x, y。
Change of base: logₐ x = log_b x / log_b a
This allows you to compute logarithms in any base using a calculator that provides only base 10 or base e logs.
此公式可利用仅有常用对数或自然对数的计算器计算任意底数的对数。
The natural logarithm: ln x = log_e x, where e ≈ 2.718
The function eˣ is its own derivative, and ln x has derivative 1/x.
函数 eˣ 的导数是其本身，ln x 的导数为 1/x。
6. Calculus | 微积分
Derivative of xⁿ: d/dx (xⁿ) = nxⁿ⁻¹
This power rule applies for any real constant n.
该幂函数求导法则适用于任意实常数 n。
Derivative of eˣ: d/dx (eˣ) = eˣ
The exponential function is unchanged by differentiation.
指数函数 eˣ 求导后保持不变。
Derivative of ln x: d/dx (ln x) = 1/x
The natural logarithm differentiates to the reciprocal function.
自然对数的导数是倒数函数。
Derivative of sin x and cos x: d/dx (sin x) = cos x; d/dx (cos x) = −sin x
Differentiation of sine gives cosine; differentiation of cosine gives negative sine (all angles in radians).
正弦函数的导数为余弦，余弦函数的导数为负正弦（角度均以弧度为单位）。
Indefinite integral (reverse of differentiation): ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, n ≠ −1
The constant of integration C must always be included for an indefinite integral.
对于不定积分，必须加上积分常数 C。
Integral of 1/x: ∫ (1/x) dx = ln |x| + C
This special case corresponds to the logarithm function.
这一特殊情况对应的是对数函数。
7. Statistics | 统计学
Mean of ungrouped data: x̄ = Σx / n
The arithmetic mean is the sum of all data values divided by the number of values.
算术平均数是所有数据值的总和除以数据个数。
Variance and standard deviation: s² = Σ(x − x̄)²/(n − 1)
The sample variance uses divisor n − 1 to give an unbiased estimate. Standard deviation s is the positive square root of the variance.
样本方差除以 n − 1 可得到无偏估计。标准差 s 是方差的正平方根。
Probability: P(A or B) = P(A) + P(B) − P(A and B)
For mutually exclusive events, the intersection term is zero.
对于互斥事件，交事件的概率为零。
Binomial distribution: X ~ B(n, p), P(X = r) = ⁿCᵣ pʳ (1 − p)ⁿ⁻ʳ
The binomial model applies to a fixed number n of independent trials, each with success probability p.
二项分布适用于固定次数 n 的独立试验，每次试验的成功概率为 p。
Mean and variance of binomial: E(X) = np, Var(X) = np(1 − p)
These are the standard results for a binomially distributed random variable.
这是二项分布随机变量的标准结论。
8. Mechanics | 力学
Constant acceleration equations (SUVAT): v = u + at, s = ut + ½at², s = vt − ½at², v² = u² + 2as, s = ½(u + v)t
These equations apply only when acceleration a is constant. u is initial velocity, v final velocity, s displacement, t time.
仅当加速度 a 恒定时才可使用这些方程。u 为初速度，v 为末速度，s 为位移，t 为时间。
Newton’s second law: F = ma
The resultant force F (in newtons) equals mass m (kg) multiplied by acceleration a (m/s²).
合力 F（牛顿）等于质量 m（千克）乘以加速度 a（米每平方秒）。
Momentum: p = mv
Momentum is the product of mass and velocity. Impulse = change in momentum = Ft.
动量是质量与速度的乘积。冲量等于动量的变化，即 Ft。
Moment of a force: Moment = F × d
The turning effect of a force equals the force multiplied by the perpendicular distance from the pivot.
力的转动效应等于力与到支点的垂直距离的乘积。
Equilibrium: ΣF = 0, ΣM = 0
For a body in static equilibrium, the vector sum of forces and the sum of moments about any point must both be zero.
物体处于静力平衡时，所有力的矢量和以及对任一点的合力矩均为零。
9. Functions and Transformations | 函数与变换
Composite function: (f ∘ g)(x) = f(g(x))
The output of g becomes the input of f. The order matters: in general f(g(x)) ≠ g(f(x)).
函数 g 的输出成为 f 的输入。顺序很重要，通常 f(g(x)) ≠ g(f(x))。
Inverse function: f⁻¹(x) reverses the effect of f, so f(f⁻¹(x)) = x
The inverse exists only if f is one‑to‑one. Its graph is a reflection of y = f(x) in the line y = x.
仅当 f 为单射函数时反函数才存在。其图像是 y = f(x) 关于直线 y = x 的对称图像。
Graph transformations: y = f(x) + a (vertical shift); y = f(x + a) (horizontal shift); y = −f(x) (reflection in x‑axis); y = f(−x) (reflection in y‑axis); y = kf(x) (vertical stretch by factor k); y = f(kx) (horizontal stretch by factor 1/k)
These transformations can be combined. Always apply stretches and reflections before shifts when rewriting a function.
以上变换可以组合使用。在改写函数表达式时，通常先进行伸缩与对称变换，再进行平移变换。
Domain and range: The domain is the set of allowed input values; the range is the set of possible output values.
Restrictions may arise from denominators, square roots, or logarithms. Identifying the domain is the first step in sketching a function.
定义域因分母、平方根或对数等限制而产生。确定定义域是绘制函数图像的第一步。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导Cancel reply